Several issues arise with such a method. First to solve the optimisation problem, we need to use an iterative algorithm with potentially computationally-intensive operators like the Fourier Transform involved in the first term or the Wavelet Transform involved in the regularisation term \(R({\mathb {x}})\) . Second, the regularisation term is not necessarily ideal for MR images as it is fixed and not finely tuned to medical images.
While dictionary learning [1]} solves partly the second problem, it does so at an increased computation cost.
Deep learning has therefore been recently introduced [2]}, [3]} to tackle both problems, allowing for a fast inference and learning from data.
However, there is a lack of works tackling the issue of data acquired in a non-Cartesian way in the k-space.
Indeed, many different sampling strategies exist for acquiring data in the k-space.
These strategies must meet hardware kinematic constraints on the gradient system (gradient magnitude, slew rate) and exhibit different advantages (e.g. robustness to motion, maximal sampling efficiency) and limitations (e.g. sensitivity to B0 inhomogeneities, gradient imperfections).
Different explanations of how and when to use the two non-Cartesian trajectories considered here, radial and spiral, can be found in [4]}.
Importantly, these two trajectories do not acquire the data on a uniform grid in the k-space, but instead off the grid.
Hence, it is no longer possible to use the classical fast Fourier transform (FFT) as the forward operator \(F_\Omega \) . Instead we consider its Non-Uniform extension called NUFFT.
